SMaRT: Online Reusable Resource Assignment and an Application to Mediation in the Kenyan Judiciary

Technical Summary: SMaRT – Online Reusable Resource Assignment for the Kenyan Judiciary

Problem Definition
The paper addresses the challenge of assigning mediators to civil cases within the Kenyan Court-Annexed Mediation (CAM) system. This is an online resource allocation problem characterized by three specific constraints that render existing algorithms (such as standard Multi-Armed Bandits or Bandits with Knapsacks) inefficient or inapplicable:

1. Unknown and Learning-Dependent Quality: Mediator effectiveness (Value Added or VA) is not known a priori and must be learned over time. New mediators enter the system, and their performance must be estimated while balancing exploration and exploitation.
2. Soft, Dynamic Capacity Constraints: Unlike standard "Bandits with Knapsacks" (BwK) which assume strict, non-replenishing limits, mediators have a capacity that replenishes as cases are resolved. However, the system requires immediate assignment upon case arrival, creating a non-zero probability of exceeding capacity (overload). While the judiciary prefers mediators handle no more than three concurrent cases, exceeding this is a "soft" constraint rather than a hard prohibition.
3. High-Dimensional State Space and Heterogeneous Accreditation: There are over 2,000 mediators with varying accreditations across 87 court stations and 12 case types. A mediator can only be assigned to a case if they are accredited for that specific (station, case type) pair. This creates a complex bipartite matching problem where resources have overlapping but distinct feasible task sets.

The objective is to maximize the overall case resolution rate (agreement rate) while managing the trade-off between assigning the best mediators and adhering to capacity constraints to ensure equitable workload distribution.

Methodology
The authors propose SMaRT (Selecting Mediators that are Right for the Task), an algorithm that integrates econometric estimation, Bayesian learning, and quadratic programming.

1. Value Added (VA) Estimation:

   • The authors adapt a value-added model (common in education research) to estimate mediator contribution. The model regresses case resolution outcomes on case controls (type, station, time) and mediator fixed effects.
   • To handle uncertainty and enable online learning, the system maintains a Gaussian belief distribution over each mediator's VA.
   • Updates: Upon observing a case outcome, the posterior distribution is updated via moment matching. To prevent drift from explicit estimates, the system recalibrates these posteriors weekly using the full VA estimation procedure on historical data.

2. Quadratic Programming (QP) Formulation:

   • The assignment decision is modeled as a QP over a short time horizon ($T=10$).
   • Objective: Maximize the expected sum of resolution probabilities (mediator VA + base case probability) minus a quadratic penalty term for capacity violations.
   • Constraints:
     • All real arriving cases must be assigned (hard constraint).
     • "Shadow cases" (future cases sampled from a distribution) may remain unassigned to simulate future demand.
     • Soft Capacity: The constraint allows mediators to exceed capacity, but violations are captured by a slack variable $\xi_u$. The objective function penalizes $\xi_u^2$, creating a trade-off parameter $\lambda$ that controls the severity of the overload penalty.
   • This formulation allows the algorithm to "hold back" high-performing mediators for future high-value cases (opportunity cost) rather than greedily assigning them to the current case, which could lead to overload.

3. Assignment Algorithm:

   • Upon a case arrival, the system constructs a state graph including the current case, shadow cases, and eligible mediators.
   • The QP is solved to obtain fractional assignments.
   • The algorithm assigns the arriving case to the mediator with the highest fractional allocation in the QP solution.
   • Variants: The paper tests two learning variants:
     • SMaRT(E): Uses the mean of the VA belief (exploitation-focused).
     • SMaRT(S): Samples VA from the belief distribution (Thompson Sampling-style exploration).

Key Contributions

• Novel Algorithmic Framework: SMaRT is the first to combine online learning of agent quality with soft, replenishable capacity constraints in a high-dimensional, heterogeneous accreditation setting. It moves beyond the "hard constraint" assumptions of BwK and the "delayed assignment" assumptions of other reusable resource algorithms (like OM-RR-PA).
• Tunable Trade-offs: The quadratic penalty parameter $\lambda$ allows administrators to explicitly tune the system's behavior, balancing the strictness of capacity constraints against the overall case resolution rate.
• Shadow Price Interpretability: The dual variables of the QP provide "shadow prices" for mediators, offering an interpretable metric of their marginal value in different contexts, which aids in understanding the "holding back" mechanism.

Results
The authors validate SMaRT using both stylized simulations and real-world data from the Kenyan Judiciary (2016–2025, ~30,600 cases).

• Stylized Scenarios: SMaRT consistently outperforms baselines (Least Load, Greedy, Thompson Sampling). In scenarios where a high-performing mediator is shared across cells, SMaRT successfully avoids the "greedy" trap of overloading that mediator, instead distributing cases to preserve capacity for future high-value opportunities.
• Real Data Simulations (Known VA): When mediator VAs are known, SMaRT with a low $\lambda$ achieves agreement rates close to the theoretical "Upper Bound" (assigning the absolute best mediator to every case) but with significantly lower overload (OCDM) and better equity (lower Gini Index) compared to a purely greedy approach.
• Real Data Simulations (Learning VA):
  • Blank Slate Initialization: When starting with no prior knowledge, capacity constraints naturally induce sufficient exploration, making SMaRT(E) (mean-based) perform well against Greedy baselines.
  • Warm Start Initialization: When initialized with historical data, SMaRT(S) (sampling-based) significantly outperforms Greedy and TS* baselines. It successfully identifies high-performing but less-experienced mediators that Greedy approaches overlook, achieving higher agreement rates while maintaining lower overload.
• Tunability: Increasing $\lambda$ reduces overload and improves equity (lower Gini Index) at the cost of a slight decrease in agreement rates, demonstrating the system's flexibility.

Significance and Claims
The paper claims that SMaRT provides a tractable and effective solution to a complex, real-world resource allocation problem that existing literature could not directly address due to the specific combination of soft constraints, immediate assignment requirements, and learning needs.

The authors state that based on these results, they plan to deploy SMaRT as the backend for the "smart assignment" feature in the Judiciary's digital platform (Cadaster). A Randomized Controlled Trial (RCT) has been approved to test the system in practice. The significance lies not only in the algorithmic contribution but in its potential to reduce judicial backlogs, lower costs, and improve the equitable distribution of work among mediators in a public sector setting. The paper emphasizes that the design was driven by operational constraints identified through close collaboration with the Kenyan Judiciary, ensuring the algorithm is feasible for real-world deployment.